Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) $V^i_j(u)$ of this system of hydrodynamic type, the metrics $g^{ij}_1(u)$ and $g^{ij}_2(u)$, the affinor $v^i_j(u)=g_1^{is}(u)g_{2,sj}(u)$, and also the affinors $(w_{1,n})^i_j(u)$ and $(w_{2,n})^i_j(u)$ of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.